NEUTROSOPHIC MULTI-OBJECTIVE LINEAR PROGRAMMING by Mia Amalia

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Global Journal of Engineering Science and Research Management NEUTROSOPHIC MULTI-OBJECTIVE LINEAR PROGRAMMING Surapati Pramanik* *Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District – North 24 Parganas, PIN-743126, West Bengal, India DOI: 10.5281/zenodo.59949 KEYWORDS: Multi-objective programming, neutrosophic set, neutrosophic multi-objective programming, neutrosophic optimization, neutrosophic goal programming, single valued neutrosophic set.

ABSTRACT For modeling imprecise and indeterminate data for multi-objective decision making, two different methods: neutrosophic multi-objective linear/non-linear programming, neutrosophic goal programming, which have been very recently proposed in the literatuire. In many economic problems, the well-known probabilities or fuzzy solutions procedures are not suitable because they cannot deal the situation when indeterminacy inherently involves in the problem. In this case we propose a new concept in optimization problem under uncertainty and indeterminacy. It is an extension of fuzzy and intuitionistic fuzzy optimization in which the degrees of indeterminacy and falsity (rejection) of objectives and constraints are simultaneously considered together with the degrees of truth membership (satisfaction/acceptance). The drawbacks of the existing neutrosophic optimization models have been presented and new framework of multi-objective optimization in neutrosophic environment has been proposed. The essence of the proposed approach is that it is capable of dealing with indeterminacy and falsity simultaneously.

1. INTRODUCTION Multi-objective programming has evolved in the last six decades into a recognized specially of operations research. Its development has occurred primarily in three disciplines, namely, operations research, economics and psychology. In 1955, Gass and Saaty [1] studied the first approach applicable to multi-objective programming problem. Multi-objective programming and planning is concerned with decision making problem having several conflicting objectives. It is one of the popular methods to deal with decision problems. Multi-objective programming problem may be characterized by an attempt to optimize a set of potentially conflicting objectives as completely as possible in an environment comprised of a set of finite resources, conflicting interest and a set of constraints. When the objective functions and constraints are linear, the multi-objective programming problem is a linear. If any objective function and/ or constraints are nonlinear, then the problem is called as a nonlinear multi-objective programming prolem. Several computational methods have been proposed in the literature for characterizing Pareto optimal solutions depending on the different approaches to scalarize the multi-objective programming problems. The details of multi-objective programming problem can be found in the books authored by Hwang, and Masud [2], M. Zeleny [3], R.E. Steur [4], Chankong and Haimes [5], M. Sakawa [6], Lai, and Hwang [7], K. Miettinen [8], For constructing of a multi-objective linear programming (MOLP) problem, various factors related to the problem should be reflected in the description of the objective functions and the constraints. The objectives functions and constraints may be characterized by many parameters. It is, naturally, recognized that the possible values of these parameters are often imprecisely or ambiguously known to the domain experts. To deal with this uncertainty, researchers employed fuzzy set due to L.A. Zadeh [9]. In 1970, Bellman and Zadeh [10] introduced three basic concepts, namely, fuzzy goal, fuzzy constraint, and fuzzy decision and explored the application of these concepts to decision making processes under fuzziness. In 1978, H. –J. Zimmermann [11] extended his fuzzy linear programming [12] to MOLP. In 1981, H. Leberling [13] studied special nonlinear functions and showed that the resulting nonlinear programming problem can be equivalently transformed into a conventional linear programming problem. In 1981, E.L. Hannan [14] adopted piecewise linear membership function to represent the fuzzy goal of the decision maker and converted the multi-objective programming problem into the ordinary linear programming problem. In 1983, M. Sakawa [6] proposed interactive fuzzy multi-objective programming problem using five

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Global Journal of Engineering Science and Research Management types of membership functions, namely, linear, exponential, hyperbolic, hyperbolic inverse, and piecewise linear functions. In the same study, M. Sakawa [6] presented linear fuzzy multi-objective programming problem by combining the use of the bisection method and the linear programming method [15]. K. Miettinen [16] presented an overview of the interactive methods for solving non-linear multi-objective programming problem and introduced the basic features of several methods by providing some theoretical results. Various methods have been proposed in the literature for MOLP to derive a satisfactory solution for decision maker based on their subjective value judgment and preference. Some popular types of such methods are namely, goal programming [17-35], fuzzy goal programming [36-61], interactive programming [2], fuzzy programming [6266] and interactive fuzzy programming [6]. Many optimization approaches/methods and techniques for modeling and solving fuzzy MOLP problems have been proposed in the literature to deal with decision making situations, which involve fuzzy values in objective functions, parameters, constraints, or goals. However, in the fuzzy model, the degree of non-membership (falsity/ rejection) as independent component due to K. Atanassov [67, 68] has not been incorporated. The principles of fuzzy optimization problems have been critically studied by P. Angellov and proposed intuitionistic fuzzy programming [69, 70] by considering truth membership (acceptance) and falsity (nonmembership/rejection) simultaneously. P. Angellov [71] also presented multi-objective optimization in air– conditioning systems based on intuitionistic fuzzy programming method. Thereafter few studies [72-90] have been reported in the literature. In 2005, Pramanik and Roy [91] presented intuitionistic fuzzy goal programming by extending fuzzy goal programming model of Pramanik and Roy [47] in intuitionistic fuzzy environment. Pramanik and Roy [92, 93] also presented intuitionistic fuzzy goal programming to transportation problem and quality control problem respectively. Multi objective programming in crisp and fuzzy environment have been well developed in order to deal realistic problems. Multi-objective programming in intuitionistic fuzzy environment is still in its infancy. MOLP in fuzzy and intuitionistic fuzzy environment are not capable of dealing with indeterminacy which exists in realistic multi-objective programming problem. So to deal MOLP involving indeterminacy, neutrosophic set studied by F. Smarandache [94, 95, 96, 97] and single valued neutrosophic set [98] are suitable tools. In 2015, Roy and Das [99] proposed neutrosophic optimization approach to solve multi-objective linear programming problem that can be considered as an extension of fuzzy programming [11] and intuitionistic fuzzy optimization [70]. In 2015, Das and Roy [100] proposed multi-objective non-linear programming problem based on neutrosophic optimization technique. Hezam et al. [101] presented Taylor series approximation to solve neutrosophic multi-objective programming problem. Kar et al. [102] applied single valued neutrosophic set theory to generalized assignment problem. Kar et al. [103] also presented neutrosophicmulti-criteria assignment problem. In 2015, Kour and Basu [104] presented neutrosophicreal life transportation problem. In 2016, Thamaraiselvi and Santhi [105] presented neutrosophgic transportation model. In the optimum solution, Thamaraiselvi and Santhi [105] considered that the degrees of indeterminacy and falsity are the same in the optimum level. In 2016, Abdel-Baset et al. [106] presented two models of neutrosophic goal programming. Roy and Das [107] applied neutrosophic goal programming model of Abdel-Baset et al. [106] to bank investment problem. In 2016, S. Pramanik [108] critically studied the results of neutrosophic optimization models presented in [99, 100,101, 106] and presented new direction of research and proposed new framework of neutrosophic linear goal programming. In Pramanik’s model [107] falsity membership function and indeterminacy membership functions are minimized while truth membership functions are maximized. Neutrosophic optimization is an open field for research work. Very little research work has been reported on neutrosophic optimization in the literature. Since in the studies [99, 100], the indeterminacy membership function has been maximized and that is not realistic goal of an organization, new methodology is urgently needed to address the issue. In this paper, new approach to neutrosophic multi-objective programming has been presented by extending Zimmermann’s approach [11] in neutrosophic environment. New insight in neutrosophic multi-objective programming has been also introduced by providing the concept of minimizing the indeterminacy membership function in multi-objective linear programming problem.

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Global Journal of Engineering Science and Research Management Remainder of the paper has been organized in the following way. Section 2 presents some basic definitions of neutrosophic sets. Section 3 is devoted to present the proposed framework of neutrosophic multi-objective programming problems. Section 4 presents the conclusion and future direction of research.

2. PRELIMINARIES We recall some basic definitions related to neutrosophic sets which are important to develop the paper. 2.1 Definition: Neutrosophic set [94] Let Y be a space of points (objects) with a generic element y  Y. A neutrosophic set S in Y is characterized by a truth membership function TS ( y) , an indeterminacy membership function I S ( y) , and a falsity membership function FS ( y) and is denoted by S {y, TS ( y), IS ( y), FS ( y) y  Y} Here TS ( y) , I S ( y) and FS ( y) can be defined as follows: 

TS : Y →]

IS : Y→]



FS : Y →]

0, 1+ [



0, 1+ [

Here, TS ( y) , I S ( y) and FS ( y) are the real standard and non-standard subset of ]  0, 1+ [ . In general, there is no restriction on TS ( y) , I S ( y) , and FS ( y) . Therefore, 

0 ≤ Inf TS ( y) + inf I S ( y) + inf FS ( y) ) ≤ Sup TS ( y) + Sup I S ( y) +Sup FS ( y) ≤ 3+ 2.2Definition: Single valued neutrosophic set [98] Let Y be a space of points with generic element yY. A single valued neutrosophic set S in Y is characterized by a truth-membership function TS(y), an indeterminacy-membership function IS (y) and a falsity-membership function FS(y), for each point y in Y, TS ( y) , I S ( y) , FS ( y) .[0, 1]. When Y is continuous, then single-valued neutrosophic set S can be written as S =   TS ( y), I S ( y), FS ( y)  / y, y  Y. Y

When Y is discrete, single-valued neutrosophic set S can be written as follows: n

S =   TS ( y i ), IS ( y i ), FS ( y i )  yi, yiY i 1

2.3 Definition: Complement of a single valued neutrosophic set [98] The complement of a single valued neutrosophic set S is denoted by S c and is defined by T c ( y)  FS ( y) ; I c ( y)  1 IS ( y) ; F c ( y)  TS ( y) S S S

2.4 Definition: Equality of two single valued neutrosophic sets [98] Equality of two single valued neutrosophic sets M and N are equal, written as M= N, if and only if M ⊆ N and M  N. 2.5. Definition: Union of two single valued neutrosophic sets [109] The union of two single valued neutrosophic sets M and N is a single valued neutrosophic set P, written as P = M  N, whose truth membership, indeterminacy-membership and falsity membership functions are related to those of M and N by TP ( y)  max ( TM ( y), TN ( y)) ; I P ( y)  min ( I M ( y), I N ( y)) ; FP ( y)  min ( FM ( y), FN ( y)) for all y in Y. 2.6. Definition: Intersection of two single valued neutrosophic sets [109] The intersection of two single valued neutrosophic sets M and N is a neutrosophic set P written as P = M  N, whose truth membership, indeterminacy-membership and falsity membership functions are related to those of M and N by TP ( y)  min ( TM ( y), TN ( y)) ; I p ( y)  max ( I M ( y), I N ( y)) ; FP ( y)  max ( FM ( y), FN ( y)) for all y in Y. Definition 2.7 [109]: Assume that { M t : tT} be an arbitrary family of single valued neutrosophic sets in Y, then i)  M t can be defined as follows: t T

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Global Journal of Engineering Science and Research Management  M t = y,  TM t ( y),  I M t ( y),  FM t ( t )

t T

(ii)  M t can be defined as follows: t T

 M t = y,  TM t ( y),  I M t ( y),  FM t ( t )

t T

3. FORMULATION OF NEUTROSOPHIC MULTI-OBJECTIVE LINEAR PROGRAMMING To formulate neutrosophic programming, we start from multi-objective programing problem in crisp environment. Consider an optimization problem of the form in crisp environment: Max  j ( y) , j = 1, 2, …,q1 (1) Subject to  j ( y) ≤ 0, j = q1+1, …, q

y0 where  j ( y) represents the i-th objective function, y is the vector of n decision variables (y1, y2, …, yn ),  j ( y) denotes i-th constraint, q1 denotes the number of objective functions and q- q1 denotes the number of constraints. 3.1 Analogous fuzzy optimization problem In general, fuzzy optimization problem comprises of a set of objectives and constraints. The objectives and or constraints or parameters and relations can be expressed by fuzzy sets, which explain the degree of satisfaction of the respective conditions and expressed by their membership functions [11]. Consider the analogous fuzzy optimization problem: ~

Max  j ( y) , j = 1, 2, …, q1 Subject to

(2)

 j ( y)  0, j = q1+1, …, q

y0 ~

 ax denotes fuzzy maximization and  denotes the fuzzy inequality. To maximize the degree of membership of the objectives and constraints to the respective fuzzy sets, we can write: Max j( y ), y n, j = 1, 2, …, q1, q1+1, …, q (3) Subject to 0 i( y )  1, j = 1, 2, …, q1, q1+1, …, q. Here j( y ) denotes the membership function of the j-th objective function  j ( y) ( j = 1, 2, …q1) and j( y ) denotes the membership function of the j-th membership function of the constraint  j ( y) (j = q1+1, …, q). Minimum operator of Bellman and Zadeh [10] can be applied to the optimization problem (3). q

 D( y ) =   j ( y) , y  0 , j = 1, 2, …, q1, q1+1, …, q

(4)

Therefore,  D( y )   j ( y) , j = 1, 2, …, q1, q1+1, …, q.

(5)

j1

According to Zimmermann [11], the problem can be solved as follows: o

 D ( v ) = Max (min ( 1 ( y) ,  2 ( y) , …,  q1 ( y) ,  q11 ( y) , …,  q ( y) ) Subject to

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(6)

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Global Journal of Engineering Science and Research Management 0 j( y )  1, j = 1, 2, …, q1, q1+1, …, q,

y0 . The problem (6) is equivalent to the following problem: Max  Subject to  j( y ), j = 1, 2, …, q1, q1+1, …, q,

(7)

y0 . 3.2 An analogous intuitionistic fuzzy optimization (IFO) problem An analogous intuitionistic fuzzy optimization problem can be represented as follows: To maximize the degree of acceptance of intuitionistic fuzzy objective functions and constraints, and to minimize the degree of rejection of intuitionistic fuzzy objective functions and constraints we can write: Max j( y ), y n , j = 1, 2, …, q1, q1+1, …, q (8) Min j( y ), y n , j = 1, 2, …, q1, q1+1, …, q Subject to j( y )+ j( y )  1, j = 1, 2, …, q1, q1+1, …, q,

(9)

0 j( y )  1, j = 1, 2, …, q1, q1+1, …, q, 0 j( y )  1, j = 1, 2, …, q1, q1+1, …, q,

y0 . Here, j( y ) denotes the degree of membership of j-th objective function  j ( y) (j = 1, 2, …q1) and j( y ) denotes the degree of j-th membership function of constraint  j ( y) (j = q1+1, …, q). Here, j( y ) denotes the degree of non-membership of j-th objective function  j ( y) (j = 1, 2, …q1) and j( y ) denotes the degree of j-th non-membership function of constraint  j ( y) (j = q1+1, …, q). Conjunction [68] of two intuitionistic fuzzy sets A and B can be defined as follows: A B= { y , A( y )  B( y ),A( y )  B( y ) y n}, (10) where A represents an intuitionistic fuzzy objectives and B represents intuitionistic fuzzy constraints. This conjunction operator can be easily generalized and applied to the IFO problem. q

j1

Here, D = { y , D( y )),D( y ) y n}, D( y ) =  j( y ),D( y ) =  j( y )

(11)

Here D represents an intuitionistic fuzzy set based representation of the decision. Min-operator can be used for conjunction and max-operator for disjunction [70]. q

D( y ) =  j( y ), y n, j = 1, 2, …, q1, q1+1, …, q, j1

(12)

D( y ) =  j( y ), y n, j = 1, 2, …, q1, q1+1, …, q,

(13)

Therefore, D( y ) j( y ),D ( y ) j( y ), j = 1, 2, …, q1, q1+1, …, q. Therefore, for optimal decision we can write Max  , Min  , Subject to j( y ) , j = 1, 2, …, q1, q1+1, …, q,

(14)

j1

j( y ) , j = 1, 2, …, q1, q1+1, …, q, 0    1, j = 1, 2, …, q1, q1+1, …, q,

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Global Journal of Engineering Science and Research Management 0    1, j = 1, 2, …, q1, q1+1, …, q,  +  1,

y0 . Here  represents minimal acceptable degree of objectives and constraints and  represents the maximal degree of rejection of objectives and constraints. Now the IFO problem (7) can be transformed into the following crisp (non-fuzzy) optimization problem: Max ( ) (15) Subject to j( y ) , j = 1, 2, …, q1, q1+1, …, q, j(( y ) , j = 1, 2, …, q1, q1+1, …, q, 0    1, j = 1, 2, …, q1, q1+1, …, q 0    1, j = 1, 2, …, q1, q1+1, …, q  +  1,

y0 . 3. 3. Formulation of neutrosophic multi objective linear programming Neutrosophic optimization problem can be represented as follows: To maximize the degree of acceptance (truth) of neutrosophic objectives and constraints, to minimize the degree of indeterminacy and to minimize the degree of rejection (falsity) of neutrosophic objectives and constraints: Max j( y ), y n, j = 1, 2, …, q1, q1+1, …, q, (16) Min i( y )[0, 1], j = 1, 2, …, q1, q1+1, …, q,

(17)

Min j( y ), y n, j = 1, 2, …, q1, q1+1, …, q , Subject to 0 j ( y )  1, j = 1, 2, …, q1, q1+1, …, q,

(18)

0 j ( y ) 1, j = 1, 2, …, q1, q1+1, …, q, 0 j( y )  1, j = 1, 2, …, q1, q1+1, …, q, j( y ) + i( v )+ j( y )  3, j = 1, 2, …, q1, q1+1, …, q,

y0 . Here j( y ) denotes the degree of truth membership of y to the j-th SVNS, j( y ) denotes the degree of indeterminacy and j( y ) denotes the degree of falsity (rejection) of functions y from the j-th SVNS. Conjunction [109] of SVNSs is defined by G C = { y , G( y )  C( y ),G( y )  C( y ), G( y )  C( y ) y n} (19) Here G represents a neutrosophic objective function and C represents neutrosophic constraint. This conjunction operator can be easily generalized and applied to the neutrosophic optimization problem: q

j 1

j1

D = { y , D( y )),D( y ) v n}, D( y ) =  j( y ),D( y )=  j( y ) , D( y ) =  j( y )

(20)

Here D represents a single valued neutrosophic set based representation of the decision. Min-operator has been used for conjunction and max-operator for disjunction: q

j1

D( y ) =  j ( y ), y n, D( y ) =  j( y ), y n, D( y ) =  j ( y ), y n.

(21)

Therefore, D( y ) j( y ),D( y ) j( y ), D( y ) j( y ),, j = 1, 2, …, q1, q1+1, …, q.

(22)

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Global Journal of Engineering Science and Research Management Here j( y ) denotes the degree of truth membership of y to the j-th SVNS, j( y ) denotes the degree of indeterminacy membership, and i( y ) denotes the degree of falsity (rejection) of functions y from the j-th SVNS. Neutrosophic multi-objective linear programming can be presented as follows: Max (      ) Subject to j( y ) , j = 1, 2, …, q1, q1+1, …, q,

(23)

j( y )   , j = 1, 2, …, q1, q1+1, …, q, j(( y ) , j = 1, 2, …, q1, q1+1, …, q, 0    1, j = 1, 2, …, q1, q1+1, …, q, 0    1, j = 1, 2, …, q1, q1+1, …, q, 0    1, j = 1, 2, …, q1, q1+1, …, q,  +  + 3,

y  0. To solve this problem, indeterminacy membership functions and falsity membership functions should be suitably constructed in the decision making context.

CONCLUSION This paper deals with the framework of neutrosophic multi-objective linear programming problem. The essence of the proposed neutrosophic multi-objective linear programming problem is that it is capable of dealing with indeterminacy and falsity simultaneously. Roy and Das [99] and Das and Roy [100] presented neutrosophic multi-objective linear programming and neutrosophic multi-objective non-linear programming respectively. However, in their approaches they maximize indeterminacy membership function that is not realistic in decision making context. In this paper the definition of intersection of two single valued neutrosophic sets due to Salama and Alblowi [109] has been utilized and minimization of falsity membership function and indeterminacy membership functions have been simultaneously considered. The proposed framework of neutrosophic multiobjective programming problem reflects the new direction of research in neutrosophic environment. Optimization problem in neutrosophic environment is a promising field of study. Neutrosophic optimization problem does, however, need a broader philosophy and new methods of dealing with problems in more versatile ways. To draw attentions, neutrosophic optimization technique must open its eyes to fresh possibilities dealing with, clearly defined indeterminacy function and falsity membership function simultaneously in realistic way. The author hopes that the proposed framework of neutrosophic multi-objective linear programming will accelerate the study of optimization problem in neutrosopohic environment.

ACKNOWLEDGEMENTS The author is thankful to Prof. (Dr.) FlorentinSmarandache, Mathematics & Science Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA, for his encouragement for the present research.

REFERENCES 1. 2. 3. 4. 5.

Gass, S., and Saaty, T.1955. The computational algorithm for the parametric objective function. Noval Research Logistics Quarterly2, 39-45. Hwang, L. and Masud, A. S. M. 1979. Multiple objective decision making – methods and applications, Berlin Springer-Verlag. Zeleny, M. 1982. Multiple criteria decision making. McGraw-Hill, New York. Steuer, R.E. 1986. Multiple criteria optimization: theory, computation, and application. John Wiley &Sons, New York. Chankong, V. and Haimes, Y.Y. 1983. Multiobjective decision making: theory and methodology. North Holland, Amsterdam.

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Global Journal of Engineering Science and Research Management 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

17.

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

31. 32. 33. 34. 35.

36. 37.

Sakawa, M.1993. Fuzzy sets and interactive multiobjective optimization. Plenum Press, New York. Lai, Y. -J. and Hwang, C. -L. 1994. Fuzzy multiple objective decision making. Springer Verlag, Berlin. Miettinen, K. 1999. Nonlinear multiobjective optimization, Kluwer Academic Pub., Boston Zadeh, L.A. 1965. Fuzzy Sets. Information and Control 8, 338353. Bellman, R.F., and Zadeh, L. A. 1970. Decision-making in a fuzzy environment. Management Sciences17, 141164. Zimmermann, H. J. 1978. Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1, 4555. Zimmermann, H. J. 1976. Description and optimization of fuzzy systems. International Journal of General Systems 2, 209-215. Liberling, H. 1981.On finding compromise solution in multicriteria problems using the fuzzy minoperator. Fuzzy Sets and Systems 6, 105-228. Hannan, E. L. 1981. Linear programming with multiple fuzzy goals. Fuzzy Sets and Systems 6, 235248. Dantzig, G.B.1961. Linear programming and extensions. Princeton University Press, New Jersy. Miettinen, K. 2002. Interactive nonlinear multiobjective procedures in: M. Ehrgott, and X. Gandibleux (eds.) Multiple criteria optimization: state of the art annotated bibliographic surveys, Kluwer Academic Publishers, Boston, 227-276. Jones, D.F., and Tamiz, M. 2002. Goal programming in the period 1990-2000, in: M. Ehrgott and X.Gandibleux (eds.), Multiple criteria optimization: state of the art annotated bibliographic surveys, Kluwer 129-170. Charnes, A., and Cooper, W.W. 1961. Management models and industrial applications of linear programming. Wiley, New Work. Ijiri, Y. 1965. Management goals and accounting for control.North–Holland, Amsterdam. Lee, S. M. 1972. Goal programming for decision analysis.Auerbach Publishers, Philadelphia. Ignizio, J.P. 1976. Goal programming and extensions. Lexington, Massachusetts, D. C. Health. Schniederjans, M. J. 1984. Linear goal programming. Potrocelli Books, New Jersey. Olson, D. L.1984. Comparison of four goal programming algorithms. Journal of the Operational Research Society. 19 (3), 373 – 396. Romero, C. 1986. A survey of generalized goal programming. European Journal of Operational Research, 25(2). pp. 183 – 191. Romero, C. 1991. Handbook of critical issues in goal programming.Pergamon Press, Oxford. Tamiz, M., and Jones, D.F. Goal programming: recent developments in theory and practice. International Journal of Management and Systems 14 (1), 1- 16. Schniederjans, M. J. 1995. Goal programming: methodology and applications, Kluwer Academic Publishers, Boston. Tamiz, M., Jones, D. F., and El-Darzi, E. 1994. A review of goal programming and its applications. Annals of Operations Research 58, 39 – 53. Tamiz, M., Jones, D.F., and Romero, C. 1998. Goal programming for decision making: an overview of the current state-of-the-art. European Journal of Operational Research 111, 567–581. Romero, C., Tamiz, M., and Jones, D. F.1998. Goal programming, compromise programming and reference point method formulations: linkages and utility interpretations. Journal of the Operational Research Society 49, 986 – 991. Romero, C.2001. Extended lexicographic goal programming: a unifying approach, Omega 29, 63–71. Romero, C. 2004, A general structure of achievement function for a goal programming model. European Journal of Operational Research 153, 675 – 686. Chang, C.-T. 2007. Multi-choice goal programming. Omega 35, 389–396. Chang, C.-T. 2008. Revised multi-choice goal programming. Applied Mathematical Modelling 32, 2587–2595. Banerjee, D., Pramanik, S. 2012. Goal programming approach to chance constrained multi-objective linear fractional programming problem based on Taylor’s series approximation. International Journal of Computers & Technology, 2(2), 77-80 Narasimhan, R. 1980. Goal programming in a fuzzy environment. Decision Sciences 11, 325-336. Hannan, E. L. 1981. On fuzzy goal programming. Decision Science 12 (3), 522-531.

http: // www.gjesrm.com

[Pramanik et al., 3(8): August, 2016]

ISSN 2349-4506 Impact Factor: 2.785

Global Journal of Engineering Science and Research Management 38. Ignizio, J. P. 1982. On the (re) discovery of fuzzy goal programming. Decision Sciences13, 331336. 39. Tiwari, R. N., Dharma, S., and Rao, J.R. 1986. Priority structure in fuzzy goal programming. Fuzzy Sets and Systems 19, 251–259. 40. Tiwari, R. N., Dharma, S., and Rao, J.R. 1987. Fuzzy goal programming- an additive model. Fuzzy Sets and Systems24, 27 – 34. 41. Mohamed, R.H. 1992. A chance constrained fuzzy goal program. Fuzzy Sets and Systems 47 (2), 183186. 42. Mohamed, R.H. 1997. The relationship between goal programming and fuzzy programming. Fuzzy Sets and Systems89, 215–222. 43. Ramik, J. 2000. Fuzzy goals and fuzzy alternatives in goal programming problems. Fuzzy Sets and Systems111, 81 – 86. 44. Pal, B. B., and Moitra, B.N. 2003. A goal programming procedure for solving problems with multiple fuzzy goals using dynamic programming.European Journal of Operational Research 144(3) 480 – 491. 45. Pramanik, S., and Roy, T.K. 2005. A fuzzy goal programming approach for multi-objective capacitated transportation problem. Tamsui Oxford Journal of Management Sciences, 21(1), 75-88, ISSN no: 02585375. 46. Pramanik, S., and Roy, T.K. 2006. A fuzzy goal programming technique for solving multi-objective transportation problem. Tamsui Oxford Journal of Management Sciences, 22 (1), 67-89. 47. Pramanik, S., and Roy, T.K. 2007. A fuzzy goal programming approach for multilevel programming problems. European Journal of Operational Research 176 (2), 1151–1166. 48. Pramanik, S., and Roy, T.K. 2008. Multi-objective transportation model with fuzzy parameters: a priority based fuzzy goal programming. Journal of Transportation Systems Engineering and Information Technology 8 (3) 40-48. 49. S. Pramanik, Dey, P.P., and Giri, B.C. 2011. Decentralized bilevelmultiobjective programming problem with fuzzy parameters based on fuzzy goal programming. Bulletin of Calcutta Mathematical Society 103 (5), 381-390. 50. Pramanik, S., and Biswas, P. 2011. Priority based fuzzy goal programming method for solving multiobjective assignment problem with fuzzy parameters. International Journal of Mathematics and Computational Methods in Science & Technology 1(6), 14-26. 51. Pramanik, S., and Dey, P.P. 2011. Quadratic bi-level programming problem based on fuzzy goal programming approach. International Journal of Software Engineering & Applications 2(4), 41-59. 52. Pramanik, S., and Dey, P.P. 2011. Bi-level multi-objective programming problem with fuzzy parameters. International Journal of Computer Applications, 30 (10) 13-20. 53. Pramanik, S. 2012. Bilevel programming problem with fuzzy parameter: a fuzzy goal programming approach. Journal of Applied Quantitative Methods. 7(1), 09-24. 54. Pramanik, S., and Banerjee, D. 2012. Multi-objective chance constrained capacitated transportation problem based on fuzzy goal programming, International Journal of Computer Applications 44(20), 42-46. 55. Pramanik, S., and Banerjee, D. 2012. Chance constrained multi-objective linear plus linear fractional programming problem based on Taylor’s series approximation. International Journal of Engineering Research and Development 1(3), 55-62. 56. Pramanik, S., and Biswas, P. 2012. Multi-objective assignment problem with generalized trapezoidal fuzzy numbers. International Journal of Applied Information Systems 2 (6), 13-20. 57. Pramanik, S., Dey, P.P., and Roy, T.K. 2012. Fuzzy goal programming approach to linear fractional bilevel decentralized programming problem based on Taylor series approximation. The Journal of Fuzzy Mathematics, (1), 23120- 238. 58. Dey, P.P., Pramanik, S., Giri, B.C. 2014. TOPSIS approach to linear fractional bi-level MODM problem based on fuzzy goal programming. Journal of Industrial and Engineering International 10(4), 173-184. 59. Pramanik, S., Banerjee, D., and Giri, B.C. 2015. Multi-level multi-objective linear plus linear fractional programming problem based on FGP approach. International Journal of Innovative Science Engineering and Technology, 2 (6), 153-160. 60. Pramanik, S. 2015. Multilevel programming problems with fuzzy parameters: a fuzzy goal programming approach. International Journal of Computer Applications 122(21), 34-41.

http: // www.gjesrm.com

[Pramanik et al., 3(8): August, 2016]

ISSN 2349-4506 Impact Factor: 2.785

Global Journal of Engineering Science and Research Management 61. Banerjee, D., Mondal, K., and Pramanik, S. 2016. Fuzzy goal programming approach for soil allocation problem in brick-fields-a case study. Global Journal of Engineering Science and Research Management 3(3), 1-16. 62. Luhandjula, M.K. 1987. Multiple objective programming problem with possibilistic coefficients. Fuzzy Sets and Systems 21, 135145. 63. Rommelfanger, H. 1989. Interactive decision making in fuzzy linear optimization problems. European Journal of Operational Research 41, 210-217. 64. Ramik, J., and Rommelfanger, H. 1993. A single and a multi-valued order on fuzzy numbers and its use in linear programming with fuzzy coefficients. Fuzzy Sets and Systems86, 299 306. 65. Carlsson. C., and Fuller, R. 1996. Fuzzy multiple criteria decision making: recent developments. Fuzzy Sets and Systems 78, 139-152. 66. Inuiguchi, M., and Ramik, J. 2000. Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets and Systems 111, 3-28. 67. Atanassov, K. 1983. Intuitionistic fuzzy sets, in: Proceedings of the VII ITKR’s Session, Sofia (Deposed in Central Sci.-Techn. Library of Bulgaria Academy of Science), 1677-1684. 68. Atanassov, K. 1986. Intuitionistic fuzzy sets. Fuzzy Sets and Systems20 (1), 87  96. 69. Angelov, P. 1995. Intuitionistic fuzzy optimization. Notes on Intuitionistic Fuzzy Sets 1 (2), 123 129. 70. Angelov, P. 1997. Optimization in an intuitionistic fuzzy environment. Fuzzy Sets and Systems 86, 299 306. 71. Angelov, P. 2001. Multi-objective optimization in air–conditioning systems: comfort/discomfort definition by IF sets. Notes on Intuitionistic Fuzzy Sets 7(1), 10-21. 72. Jana, B., and Roy, T.K. 2007. Multi-objective intuitionistic fuzzy linear programming and its application in transportation model, Notes on Intuitionistic Fuzzy Sets 13(1), 1-18. 73. Mahapatra, G.S., Mitra, M., and Roy, T.K. 2010. Intuitionistic fuzzy multi-objective mathematical programming on reliability optimization model. International Journal of Fuzzy Systems, 12(3), 259266. 74. Pramanik, S., Dey, P.P., and Roy, T. K. 2011. Bilevel programming in an intuitionistic fuzzy environment. Journal of Technology 42, 103-114. 75. Chakrabortty, S., Pal, M., and Nayak, P.K. 2011. Intuitionistic fuzzy optimization technique for the solution of an EOQ model. Notes on Intuitionistic Fuzzy Sets, 17(2), 52-64. 76. Parvathi, R., and Malathi, C. 2012. Intuitionistic fuzzy simplex method. International Journal of Computer Applications 48 (6), 39–48. 77. Hussain, R. J., and Kumar, S. P. 2012. Algorithmic approach for solving intuitionistic fuzzy transportation problem. Applied Mathematical Sciences. 6(77-80), 3981–3989. 78. Gani, A.N. and Abbas, S. 2012. Solving intuitionstic fuzzy transportation problem using zero suffix algorithm. International Journal of Mathematics Sciences & Engineering Applications 6, 73–82. 79. Garai, A., and Roy, T.K. 2013. Optimization under generalized intuitionistic fuzzy environment. International Journal of Computer Applications 73(13), 20-23. 80. Garai, A., and Roy, T.K. 2013. Intuitionistic fuzzy optimization: usage of hesitation index. Notes on Intuitionistic Fuzzy Sets 19 (4), 60–68. 81. Chakrabortty, S., Pal, M., and Nayak, P.K. 2013. Intuitionistic fuzzy optimization technique for Pareto optimal solution of manufacturing inventory models with shortages. European Journal of Operational Research228(2), 381–387. 82. Bharati, S.K., and Singh, S.R. 2014. Solving multi-objective linear programming problems using intuitionistic fuzzy optimization: a comparative study. International Journal of Modelling and Optimization 4(1), 10-15. 83. Kalaiarasi, K. 2014. Solution of stochastic cooperative inventory models by intuitionistic fuzzy optimization technique. International Journal of Computing Science and Information Technology 2(1), 6-10. 84. Garg, H., Rani, M., Sharma, S.P., and Vishwakarma, Y. 2014. Intuitionistic fuzzy optimization technique for solving multi-objective reliability optimization problems in interval environment.Expert Systems with Applications 41(7), 3157–3167.

http: // www.gjesrm.com

[Pramanik et al., 3(8): August, 2016]

ISSN 2349-4506 Impact Factor: 2.785

Global Journal of Engineering Science and Research Management 85. Dey, S., and Roy, T.K. 2014. Optimized solution of two bar truss design using intuitionistic fuzzy optimization technique.I.J. Information Engineering and Electronic Business4 , 45-51. 86. Bharati, S.K., and Singh, S.R. 2014. Intuitionistic fuzzy optimization technique in agricultural production planning: a small farm holder perspective. International Journal of Computer Applications 89 (6), 17-23. 87. Bharati, S.K., and Singh, S.R. 2015. A note on solving a fully intuitionistic fuzzy linear programming problem based on sign distance.International Journal of Computer Applications 119 (23), 30-35. 88. Garai, A., Mandal, P, and Roy, T.K. 2015. Intuitionistic fuzzy T-sets based solution technique for multiple objective linear programming problems under imprecise environment. Notes on Intuitionistic Fuzzy Sets 21(4). 89. Dey, S., and Roy, T.K. 2015. Intuitionistic fuzzy goal programming technique for solving non-linear multi-objective structural problem. Journal of Fuzzy Set Valued Analysis 2015(3), 179-193. 90. Garai, A., Mandal, P, Roy, T.K. 2016. Interactive intuitionistic fuzzy technique in multi-objective optimisation.International Journal of Fuzzy Computation and Modelling 2(1), 14-26. 91. Pramanik, S., and Roy, T.K. 2005. An intuitionistic fuzzy goal programming approach to vector optimization problem. Notes on Intuitionistic Fuzzy Sets 11(5), 01–14. 92. Pramanik, S., and Roy, T.K. 2007. An intuitionistic fuzzy goal programming approach for a quality control problem: a case study. Tamsui Oxford Journal of Management Sciences 23(3), 01–18. 93. Pramanik, S., and Roy, T.K. 2007. Intuitionistic fuzzy goal programming and its application in solving multi-objective transportation problem. Tamsui Oxford Journal of Management Sciences 23(1), 01–16. 94. Smarandache, F. 1998. A unifying field in logics: neutrosophic logic, neutrosophy, neutrosophic set, neutrosophic probability. American Research Press, Rehoboth. 95. Smarandache, F. 2002. A unifying field in logics: neutrosophic logics. Multiple Valued Logic 8 (3), 385-438. 96. Smarandache, F. 2005. Neutrosophic set. A generalization of intuitionistic fuzzy set. Internal Journal of Pure and Applied Mathematics 24, 287-297. 97. Smarandache, F. 2010. Neutrosophic set – a generalization of intuitionistic fuzzy set. Journal of Defense Resources Management 1(1), 107-116. 98. Wang, H., Smarandache, F., Zhang, Y., and Sunderraman, R. 2010. Single valued neutrosophic sets. Multisspace and Multistructure 4, 410-413. 99. Roy, R., and Das, P. 2015. A multi-objective production planning roblem based on neutrosophic linear rogramming approach. Internal Journal of Fuzzy Mathematical Archive 8(2) 81-91. 100. Das, P., and Roy, T.K. 2015. Multi-objective non-linear programming problem based on neutrosophic optimization technique and its application in riser design problem. Neutrosophc Sets and Systems 9, 8895. 101. Hezam, I.M., Abdel-Baset, M., and Smarandache, F. 2015. Taylor series approximation to solve neutrosophic Multi-objective programming problem. Neutrosophic Sets and Systems 10, 39-45. 102. Kar, S., Basu, K., and Mukherjee, S. 2015. Application of neutrosophic set theory in generalized assignment problem.Neutrosophic Sets and Systems 9, 75-79. 103. Kar, S., Basu, K., and Mukherjee, S. 2015. Solution of multi-criteria assignment problem using neutrosophic set theory. Neutrosophic Sets and Systems 11, 31-38. 104. Kour, D., and Basu, K. 2015.Application of extended fuzzy programming technique to a real life transportation problem in neutrosophic environment. Neutrosophic Sets and Systems 10, 75-88. 105. Thamaraiselvi, A., and Santhi, R. 2016.A new approach for optimization of real life transportation problem in neutrosophic environment. Mathematical Problems in Engineering. http://dx.doi.org/10.1155/2016/5950747. 106. Abdel-Baset, M., Hezam, I.M., and Smarandache, F. 2016. Neutrosophic goal programming, Neutrosophic Sets and Systems 11, 112-118. 107. Roy, R., and Das, P. 2016. Neutrosophic goal programming applied to bank three investment problem. Neutrosophic Sets and Systems 12, 97-104. 108. Pramanik, S. 2016. Neutrosophic linear goal programming. Global Journal of Engineering Science and Research Management 3(7), 01-11. 109. Salama, A.A., and Alblowi, S.A. 2012. Neutrosophic set and neutrosophic topological spaces. IOSR Journal of Mathematics (IOSR-JM) 3(4), 31-35.

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